Flow and flow rate... Halp! I'm so confused... 
I think I got the meaning of flux, it's a scalar that indicates the "quantity of a vector field (of field lines)" that crosses a surface of a given area.
So no time relation implied right? 
First: is flow and flux the same thing in English when talking about physics? (in Italian we just refer to it with the word "flusso")
Why I often read, about fluid flow (flux?) :
it's the quantity that measures volume that crosses a surface per unit time? 
Is this a misunderstanding? is this the definition of flow rate?
Is flow rate and flow the same thing in the field of velocity of a fluid? 
Is the mass ( or volume) of the fluid bond to the vectorial field in a fluid, or I can just pick the velocity of a point in the fluid without considering the mass? 
Is flow rate the dual of current for an electromagnetic field? 
How is current related to flux?
Maybe the key is the divergence theorem... I need some sleep +.+
It's a lot of questions but they are strictly related to each other, I guess, the point is that I studied those things separately, I can't build an organic connection in my head.
 A: This is largely an issue of units - flux is a general term that applies to any vector field, even those that are typically not seen as a rate. For instance, the flux of a magnetic field through a surface is a useful quantity. You might generally think of it as referring to a mathematical operation: take a vector field and a surface (with some orientation), and integrate the magnitude of the field normal to the surface across the surface. You can sort of think of this as "how much field crosses the surface", although this is imprecise - there's no material in a magnetic field that is truly crossing a surface, but that flux turns out to be an notable quantity (due to Gauss' law for magnetism or Faraday's law of induction).
Flow rate is more specific: it really is how much of something (usually a fluid) crosses a surface in a given time. It just so happens that this quantity is given as the flux of a field through a surface - for instance, if you want the volume flow rate, you just integrate the velocity field of the fluid over the surface in question. If you want the mass flow rate, you would need to integrate with respect to mass (which is the same as multiplying the velocity field by the mass density of the fluid to get a field of momentum density instead of velocity).
Note that in the examples of flow rate, the dependence with time comes from the units of the field, not from the integration. In particular, integrating over a surface (with respect to surface area) is just going to multiply whatever units you started with by a unit like meters squared - so if you started with velocity, you get volume per time and if you start with momentum density, you get mass per time - but the "per time" was already part of the units being fed into the calculation of flux.
A: In the literary sense, "Flux" is derived from the Latin word "Fluxus" which literally means flow. Mathematically, Any Integral of the type $$I = \int\int_S \mathbf{F}(x,y,z)  \cdot \mathbf{\hat{n}}\quad dS $$
is called the "flux" of $\textbf{F}$ through S.
In order to give a more physical feeling, the fluid flow is the most canonical example of flux. Note that there is no inherent temporal nature to the mathematically meaning of flux. Flux described for fluid has a time factor in it supplied by the fluid velocity field but it is not the case in general. 
In the fluid case, 
$$\text{Flux through S} = \int\int_s \rho(x,y,z)\mathbf{v}(x,y,z)\cdot \mathbf{\hat{n}} \quad dS$$
Here, $\mathbf{F} = \rho(x,y,z)\mathbf{v}(x,y,z)$, and we are talking about flux of $\mathbf{F}$. But "flux" is always not a physically intuitive quantity, an equation that takes the form of $I$, is a "Flux of $\mathbf{F}$ over $S$". I would recommend you to reffer "Div, Grad, Curl- h.m schey" for any problems in understanding vector calculus, but chapter-II, page 29 for flux. 

Imagine the case where $S$ is just a plane surface of unit area and let density $\rho$ be 1, and the fluid is moving with constant velocity $v$, what is the amount of fluid that passes this surface in unit time? Now, what if this surface is hemisphere? Now, what if the velocity is not constant? Do these steps constructively and you will understand the expression of "Flux through S". So this a case where the layman flux coincides with the mathematical flux. But the mathematical flux as I mention in the very first equation, need not coincide with this. I think you need to give up the idea of flux you have from your experience with fluids and adopt the meaning as given by  @Milo Brandt in the first part of his answer, and then from there see fluid flux as a special case emerging from the definition. 
